@TechReport{Langdon97,
  author      = "W. B. Langdon and R. Poli",
  title       = {Price's Theorem and the MAX Problem},
  institution = {University of Birmingham, School of Computer Science},
  number      = {CSRP-97-4},
  month       = {January},
  year        = {1997},
  file        = {/1997/CSRP-97-04.ps.gz},
  url         = {ftp://ftp.cs.bham.ac.uk/pub/tech-reports/1997/CSRP-97-04.ps.gz},
  abstract    = {We present a detailed analysis of the evolution of GP
                 populations using the problem of finding a program which
                 returns the maximum possible value for a given terminal
                 and function set and a depth limit on the program tree
                 (known as the MAX problem). We confirm the basic message
                 of \cite{Gathercole:1996:aicrtd} that crossover together
                 with program size restrictions can be responsible for
                 premature convergence to a sub-optimal solution. We show
                 that this can happen even when the population retains a
                 high level of variety and show that in many cases
                 evolution from the sub-optimal solution to the solution is
                 possible if sufficient time is allowed. In both cases
                 theoretical models are presented and compared with actual
                 runs. 
                 
                 Experimental evidence is presented that Price's Covariance
                 and Selection Theorem can be applied to GP populations and
                 the practical effect of program size restrictions are
                 noted. Finally we show that covariance between gene
                 frequency and fitness in the first few generations can be
                 used to predict the course of GP runs.
                 },
}